A semigroup is nilpotent of degree $3$ if it has a zero, every product of $3$ elements equals the zero, and some product of $2$ elements is non-zero. It is part of the folklore of semigroup theory that almost all finite semigroups are nilpotent of degree $3$. We give formulae for the number of nilpotent semigroups of degree $3$ on a set with $n\in\mathbb{N}$ elements up to equality, isomorphism, and isomorphism or anti-isomorphism. Likewise, we give formulae for the number of nilpotent commutative semigroups on a set with $n$ elements up to equality and up to isomorphism.
@article{10_37236_2441,
author = {Andreas Distler and J. D. Mitchell},
title = {The number of nilpotent semigroups of degree 3.},
journal = {The electronic journal of combinatorics},
year = {2012},
volume = {19},
number = {2},
doi = {10.37236/2441},
zbl = {1254.20049},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2441/}
}
TY - JOUR
AU - Andreas Distler
AU - J. D. Mitchell
TI - The number of nilpotent semigroups of degree 3.
JO - The electronic journal of combinatorics
PY - 2012
VL - 19
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/2441/
DO - 10.37236/2441
ID - 10_37236_2441
ER -
%0 Journal Article
%A Andreas Distler
%A J. D. Mitchell
%T The number of nilpotent semigroups of degree 3.
%J The electronic journal of combinatorics
%D 2012
%V 19
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/2441/
%R 10.37236/2441
%F 10_37236_2441
Andreas Distler; J. D. Mitchell. The number of nilpotent semigroups of degree 3.. The electronic journal of combinatorics, Tome 19 (2012) no. 2. doi: 10.37236/2441
